A trigonometry question ...

For a story I've been working on, I postulated a terrestrial world whose axis of spin is parallel to its orbit, and whose equator and moon's orbit is perpendicular to this world's orbit.

Another postulate is that during the seasons when the sun shines on the equator, I imagine that when it is directly over the equator it gives the surface side inhabitants 12 (out of 24) hours of daylight, so when it is at 45 degrees in the sky it would only provide 6 hours of daylight.

The final postulate is that there is a space elevator, and a city in geosynchronous orbit, based on the equator. A trip to or from the orbital city takes 4 days.

My question is, how long would the station pass through the shadow of the world on days when the sun peaks at 45 degrees over the equator?

Likewise, I figure the crawler that travels up and down the elevator would spend more and more time in darkness the closer it is to the surface ... is there some kind of formula I could use to calculate the length of time the crawler is in shadow for each day? If so what is it?

Thanks for your help in this matter.

/s/

Gloriosus
the G-man Himself

Edited by G-man, 07 July 2019 - 07:52 AM.

Let me strive every moment of my life to make myself better and better, to the best of my ability, so that all may profit by it.
Let me think of the right and lend my assistance to all who may need it, with no regard for anything but justice.
Let me take what comes with a smile, without loss of courage.
Let me be considerate of my country, of my fellow citizens, and my associates in everything I say and do.
Let me do right to all, and wrong no man. -- Doc Savage

Few people want to be moderated, most people see the need for everyone else to be moderated. -- Orpheus

Let me strive every moment of my life to make myself better and better, to the best of my ability, so that all may profit by it.
Let me think of the right and lend my assistance to all who may need it, with no regard for anything but justice.
Let me take what comes with a smile, without loss of courage.
Let me be considerate of my country, of my fellow citizens, and my associates in everything I say and do.
Let me do right to all, and wrong no man. -- Doc Savage

Few people want to be moderated, most people see the need for everyone else to be moderated. -- Orpheus